Geometry Main 14 15a 15b 18 19 20a 20b 20c 20d 20e


The 5th GTSS

GEOMETRY-TOPOLOGY SUMMER SCHOOL

İstanbul Center for Mathematical Sciences - Online
July 6-18, 2020






First Week

 

TIME              SPEAKER                  TITLE
Jul 6-11
9-10:30
Roger Nakad Spin-c structures on manifolds and geometric applications
Jul 6-11
10:30-12
Giovanni Bazzoni Conformal Symplectic and Kähler Geometry
Jul 6-11
2:30-4
Vamsi Pritham Pingali The Calabi Conjectures


Second Week

 

TIME              SPEAKER                  TITLE
Jul 13-16
10:30-12
Simone Calamai Constant Scalar Curvature Kähler metric (CscK) problem:
A priori estimates
Jul 13-17
2:30-4
Buket Can Bahadır
Special Metrics in Complex Analysis
Jul 14-17
4:30-6
İzzet Coşkun
Birational Geometry of the Hilbert Schemes of points on the plane

Scientific Commitee

 

Vicente Cortés University of Hamburg, Germany
İzzet Coşkun University of Illinois at Chicago, USA
Ljudmila Kamenova Stony Brook University, USA
Lei Ni University of California at San Diego, USA
Tommaso Pacini University of Torino, Italy
Gregory Sankaran University of Bath, UK
Misha Verbitsky IMPA, Brasil

Organizing Commitee

 

Craig van Coevering Bosphorus University
İlhan İkeda Bosphorus University
Mustafa Kalafat Nesin Mathematical Village







Registration       Poster       Participants       Arrival



Information

The 5th GTSS will be held at the IMBM, İstanbul Center for Mathematical Sciences (Online)
which is established in the main (South) Campus of Boğaziçi University (Bosphorus).
The first photo on top of this webpage demonstrates the view from the venue.
There will be about 15 mini-courses of introductory nature, related to the Geometry-Topology research subjects.
In the middle of the week there is an excursion on the Bosphorus. Do not forget to get your swimsuit with yourself.



Application

Graduate students, recent Ph.D.s and under-represented minorities are especially encouraged to the summer school.
Daily expenses including bed, breakfast, lunch, dinner is around 20€.
You may stay at cheap hostels around Taksim(Nightlife) square and take the subway to the Campus easily.
We are trying to arrange accomodation on Campus as well.

Please fill out the Registration form in order to attend the "Online" summer school. Registration is free but mandatory.

Airport: İstanbul Airport - IST is the closest one. Take a bus from the airport to the 4. Levent. Then take a taxi or subway to reach to the main/south campus.
Alternatively you can use Sabiha Gökçen Airport - SAW and take a bus from there to 4. Levent. Then take a taxi or subway to reach to the main/south campus.
Navigate the link above for more detailed arrival and venue information.

Visas: Check whether you need a visa beforehand.






Abstracts



Spin-c structures on manifolds and geometric applications

These lecture series aim to give an elementary exposition on basic results about the first eigenvalue of the Dirac operator, on compact Riemannian Spin and Spin^c manifolds and their hypersurfaces. For this, we select some key ingredients which illustrate the basic objects and some of their properties as Clifford algebras, spin and spin^c groups, connections, covariant derivatives, Dirac and Twistor operators. We end by giving beautiful geometric applications: a Lawson type correspondence for constant mean curvature surfaces in some 3-dimensional Thurston geometries, extrinsic hyperspheres in manifolds with special holonomy, Alexandrov type theorems…

Textbook or/and course webpage:

1. Th. Friedrich, Dirac operator’s in Riemannian Geometry, Graduate studies in mathematics, Volume 25, American Mathematical Society, 2000

2. H.B. Lawson and M.L. Michelson, Spin Geometry, Princeton University press, Princeton, New Jersey, 1989.

3. J.P. Bourguignon, O. Hijazi, J.L. Milhorat, A. Moroianu and S. Moroianu, A Spinorial Approach to Riemannian and Conformal Geometry, Monographs in Mathematics, European Mathematical Society (June 2015) 462 pages.

Prerequisites:

Linear Algebra, Riemannian Geometry

Conformal Symplectic and Kähler Geometry

In this series of lectures we study smooth manifolds endowed with geometric structures which are conformal, on contractible open sets, to a symplectic structure or to a Kähler metric. We study the main properties of conformal symplectic and conformal Kähler manifolds, providing many examples, proving some structure theorems, showing their relation with symplectic and Kähler manifolds, as well as with manifolds with further geometric structures. We also highlight the relevance of conformal symplectic geometry in hamiltonian mechanics. We will also report on conformal analogues of geometric structures coming from special holonomy, such as G2 and Spin(7).

Program

1. Hamiltonian mechanics

2. Conformal symplectic and Kähler structures

3. Structure theorems

4. Conformal structures related to other geometric structures

References

A. Banyaga, On the geometry of locally conformal symplectic manifolds, In: Infinite dimensional Lie groups in geometry and representation theory (Washington, DC, 2000), 79{91. World Sci. Publ., River Edge, NJ, 2002.

G. Bazzoni, Locally conformally symplectic and Kähler Geometry, EMS Surv. Math. Sci. 5(1) (2018), 129-154. Available on https://arxiv.org/abs/1711.02440.

S. Dragomir, L. Ornea, Locally Conformal Kähler Geometry, Progress in Mathematics 155, Birkhäuser, 1998.

S. Ivanov, M. Parton, P. Piccinni, Locally conformal parallel G2 and Spin(7) ma- nifolds, Math. Res. Lett., 13(2-3) (2006), 167-177.

L. Ornea, M. Verbitsky LCK rank of locally conformally Kähler manifolds with potential, J.Geom. Phys (107) (2016) 92-98.

I. Vaisman, Locally Conformal symplectic manifolds, Internat. J. Math. & Math. Sci. 8(3) (1985), 521-536.

Level and Prerequisites:

This course is intended for advanced undergraduate students, and graduate students. The prerequisites are familiarity with smooth manifolds, di erential forms, and basics of Riemannian geometry.

The Calabi Conjectures

We shall state, prove, and study applications of some of Calabi’s Conjectures.

Topics to be covered are as follows: Definition of complex manifolds and holomorphic maps, Examples. Kahler metrics and examples. Holomorphic line bundles, the canonical bundle and the first Chern class. Statement of Calabi’s volume conjecture, the representability of the first Chern class conjecture, and the Kahler-Einstein conjecture. Examples and applications. If time permits, a gentle introduction to the ideas behind the proof (method of continuity, a priori estimates).

Textbook and/or References: -

Prerequisites:

Manifold theory including Riemannian metrics and the Levi-Civita connection, complex analysis, and functional analysis. Some exposure to elliptic PDE and complex geometry will be immensely helpful but is not necessary.

A priori estimates on the Constant Scalar Curvature Kähler metric (CscK) problem

In this lecture series we present the 2018 breakthrough results by XiuXiong Chen and JingRui Cheng on the analytic a priori estimates for the fourth order fully nonlinear elliptic PDE that gives constant scalar curvature Kähler metrics. Preliminarily we deliver some motivations and historical background on the problem; we also present the correspondence between the properness of Kenergy and the existence of constant scalar curvature Kähler metrics. Then we move to the review of the a priori estimates, with special emphasis on the C^0 order ones.

Textbook, References or/and course webpage:

1. An Introduction to Extremal Kähler Metrics. Gabor Szekelyhidi. American Math Soc.

2. On the constant scalar curvature Kähler metrics, apriori estimates. XiuXiong Chen and JingRui Cheng, available on Arxiv.

3. On the constant scalar curvature Kähler metrics, existence results. XiuXiong Chen and JingRui Cheng, available on Arxiv.

Prerequisites:

Riemannian Geometry, Kähler geometry

Level: Graduate

Birational Geometry of the Hilbert Schemes of points on the plane

In these 4 lectures, I will give an introduction to the birational geometry of Hilbert schemes of points in the plane.

In lecture 1, I will introduce the Hilbert scheme of points on surfaces and discuss some of its basic geometric properties.

In lecture 2, I will introduce some key notions of birational geometry, moduli spaces of higher rank vector bundles and Brill-Noether divisors.

In lecture 3, I will introduce Bridgeland stability conditions and discuss the interaction between the chamber decomposition of the Bridgeland stability manifold and the chamber decomposition of the effective cone.

In lecture 4, I will tie everything together and discuss the effective cone of moduli spaces of sheaves on the plane. The Hilbert scheme will serve as the main example.

Level:

The intended audience of these lectures are graduate students and postdoctoral researchers who are familiar with algebraic geometry at the level of Hartshorne or Griffiths and Harris.

Textbook and/or References:

These lectures will be based on joint work with Daniele Arcara, Aaron Bertram, Jack Huizenga and Matthew Woolf.

Special Metrics in Complex Analysis

1- Complex Analysis and Differential Geometry Background

2- The Poincare Metric and Schwarz Lemma

3- Curvature, Liouville’s Theorem and Other Applications

4- Normal Families and the Spherical Metric

5- The Caratheodory and the Kobayashi Metric

6- The Bergman Metric and Some Applications

7- A Word About Several Complex Variables

Level: Graduate, advanced undergraduate

Textbook or/and course webpage:

Complex Analysis - The Geometric Viewpoint by Steven G. Krantz

Prerequisites:

Complex Analysis -->

Share via Whatsapp

(*) To be confirmed
Contact: ilaydabariss@gmail.com, berkanuze@gmail.com



Activities are supported by Nesin Mathematical Village and Turkish Mathematical Society